Problem

Source: 2017 KMO Problem 6

Tags: geometry, circumcircle



In a quadrilateral $ABCD$, we have $\angle ACB = \angle ADB = 90$ and $CD < BC$. Denote $E$ as the intersection of $AC$ and $BD$, and let the perpendicular bisector of $BD$ hit $BC$ at $F$. The circle with center $F$ which passes through $B$ hits $AB$ at $P (\neq B)$ and $AC$ at $Q$. Let $M$ be the midpoint of $EP$. Prove that the circumcircle of $EPQ$ is tangent to $AB$ if and only if $B, M, Q$ are colinear.